Definition: Let F be a subset of a metric space X. F is called(adsbygoogle = window.adsbygoogle || []).push({}); closedif whenever is a sequence in F which converges to a E X, then a E F. (i.e. F contains all limits of sequences in F) Theclosureof F is the set of all limits of sequences in F.

Claim 1: F is contained in the clousre of F.

Claim 2: The closure of F is closed.

How can we prove these formally?

For claim 1, I think we have to show x E F => x E cl(F).

For claim 2, we need to prove that cl(F) contains all limits of sequences in cl(F).

But how can we prove these?

I know these are supposed to be basic facts, but my book never gives examples of how to prove these from the definitions given above...and I have no clue...

Any help is appreciated!

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Closure & Closed Sets in metric space

**Physics Forums | Science Articles, Homework Help, Discussion**